Analysis of higher-order Lotka-Volterra models: Application of S-tensors and the polynomial complementarity problem

It is known that the effect of species' density on its growth is non-additive in real ecological systems. This challenges the conventional Lotka-Volterra model, where the interactions are always pairwise and their effects are additive. To address this challenge, we introduce HOIs (Higher-Order Interactions) which are able to capture, for example, the indirect effect of one species on a second one correlating to yet a third. Towards this end, we study a general higher-order Lotka-Volterra model. We provide an existence result of a positive equilibrium for a nonhomogeneous polynomial equation system with the help of S-tensors. Afterward, by utilizing the latter result, as well as the theory of monotone systems and results from the polynomial complementarity problem, we provide comprehensive results regarding the existence, uniqueness, and stability of the corresponding equilibrium. These results can be regarded as natural extensions of many similar ones for the classical Lotka-Volterra model, especially in the case of full cooperation, competition among two factions, and pure competition. Finally, illustrative numerical examples are provided to highlight our contributions.